Quantum Field Theory II

The following sections are included:

• Contents
• Preface

In the previous semester we covered basics and general aspects of quantum field theory. In this semester we will consider some particular, the most important field theories such as non-Abelian gauge theories (also known as Yang-Mills theories). During the last six decades, Yang-Mills theory has become the cornerstone of theoretical physics. It is seemingly the only fully consistent relativistic quantum field theory in four space-time dimensions. As such, it is the underlying theoretical framework for the Standard Model of Particle Physics (a part of which is the Glashow-Weinberg-Salam, GWS) model, which was proven to be the correct theory at all currently measurable energies. For recommended primary textbooks, see [1]…

The following sections are included:

• Vertices in Yang-Mills with quarks
• Higgsing
• From Minkowski to Euclidean formulation
• Appendix 2.1

The following sections are included:

• Path Integral
  • What is path integral
  • Quantum-mechanical example: Harmonic oscillator
• Path integral quantization
• Harmonic oscillator: Green’s function
• Appendix 3.1: On Richard Feynman

The following sections are included:

• Free scalar field theory
• Calculation of the partition function
• General remark on n-point functions
• Calculation of the scalar propagator

The following sections are included:

• Scalar field (continued)
• QFT at finite temperature

The following sections are included:

• Grassmann numbers
• Calculus on Grassmann numbers
• Generalization to complex Grassmann numbers
• Exponential integrand
• Four-dimensional spinors
  • Two and three dimensions
• Appendix 6.1. On Ettore Majorana
• Appendix 6.2: Derivation of (6.34) from (6.31)
• Appendix 6.3: On Hermann Grassmann and Felix Berezin

The following sections are included:

• Dirac fields in path integrals
• Fermions at finite temperature

The following sections are included:

• Photon fields in path integral
• Appendix 8.1: On L. D. Faddeev

I will begin this lecture from rephrasing the derivation for the Abelian case (discussed in Chapter 8) in the hope that this rephrasing can facilitate the understanding of subtle points of Yang-Mills quantization. To this end, I find it helpful to pass to the Fourier-transformed formulation, assuming that the (four-dimensional) momentum space is somehow discretized…

Today we will begin a new topic. My final goal is to teach you how to calculate the coupling constant renormalization (a.k.a running coupling) in Yang-Mills theory. It will take us some time; we will start from a simple set-up (quantum electrodynamics, QED) and gradually move toward more complicated things…

The following sections are included:

• The scalar and fermion field loops
• The gauge field and ghost loops
• Appendix 11.1: On Julian Schwinger and the background field method in QCD

The following sections are included:

• Degenerate vacua: Path integral
• Path-integral quantization in the Higgsed regime
  • Abelian Higgsed theory
• Non-Abelian case

The following sections are included:

• General
• The (classical) Heisenberg model
• The Ising model
• The Ising model at the critical point
• Solution
• Appendix 13.1: On Ernest Ising

The following sections are included:

• Charge renormalization (continued)
• Running couplings, renormalization group
• Running gauge coupling. Beta function for Yang-Mills theory with matter
  • Asymptotic freedom in Yang-Mills theories
• QED, the Landau pole
• Zeros of the β functions. Conformal points
• Appendix 14.1: Screening versus antiscreening

The phase structure of non-Abelian gauge theories is richer than that of QED. In addition to three regimes we had already discussed (Coulomb, Higgs, asymptotically free fixed point), which were known already in the 1970s, Yang-Mills and other theories can exhibit confining and conformal phases, phases with or without chiral symmetry breaking, and so on…

The following sections are included:

• Anomalous dimension
• More on dimensional transmutation
  • A few words on scale transformation

The following sections are included:

• Spontaneous breaking of chiral symmetry
• Effective Lagrangians
  • Baryons. The Skyrme model
• χSB versus color and avor
• The number of Goldstone particles (gapless excitations)
• Appendix 17.1: On Maurice Lévy and Tony Skyrme

This topic is important, since the phenomenon of anomalies plays a role in a number of subtle aspects of gauge dynamics. It also provides us with one of the few tools available at strong coupling…

The following sections are included:

• Cancellation of internal anomalies in Standard Model (SM)
• Internal anomalies in SM. First generation
• Peculiarity of the SU(2) sector
• Witten’s global anomaly in SU(2)
• Other anomalies
• Appendix 19.1: Creators of the Standard Model

The following sections are included:

• “External” anomalies
• Longitudinal part of the current
• ’t Hooft matching and its physical implications
• Spontaneous breaking of the axial symmetry
• Predicting the π⁰ → 2γ decay rate
• Appendix 20.1: Gerard ’t Hooft

In QFT-I and in this course (QFT-II) we learned how to calculate one-loop corrections in the coupling constant, mostly using gauge theories as examples, namely QED and Yang-Mills theories (QCD). In the 1950s, it was realized that expansions in the coupling constant α cannot represent conventional convergent series, with a finite radius of convergence. In the vast majority of cases the emerging series are asymptotic. This means that at first, as the expansion order grows, so does its accuracy. However, if we go too far in our perturbative expansion, typically to orders higher than n = const/α, the accuracy of the series becomes worse rather than better. In other words, the expansion coefficients increase at high orders, and the perturbative series has vanishing convergence radius. Shortly, I will explain how this can happen. Right now, I explain why this is inevitable.

The following sections are included:

• A brief review of a well-known QM problem
• Instantons in QM
• Topological charge
• Instanton gas

The following sections are included:

• Appendix A: Lecture 23. Nontrivial Topology in the Space of Fields
• Appendix B: Problems
• Appendix C: Brief History of QFT

The following section is included:

• Index